For two decades, a loose-knit group that includes some of the country's wealthiest people has underwritten the political push for school choice, promoting ballot initiatives and candidates who favor competition for traditional public schools.

But when a member of this elite group was elevated to education secretary, the appointment opened a philosophical schism that now threatens to shatter the alliance, turn billionaires against each other and possibly lead some school-choice advocates to join with teachers' unions, their archenemies.

Fueling the split is the anticipation of a plan from Presi...